CALCULUS

Problem 1.

Although calculus is used extensively in space science and technology, we shall consider in this chapter just a few problems, most of which extend or amplify ideas discussed in previous chapters. Calculus is also used in the Appendix.

PROBLEM 1. Until recently it was accepted that there were three possible states in which matter could exist: solid, liquid, and gas. Under conditions that normally prevail on Earth, these are the only states in which matter is found. However, it is now known that if the temperature is very high or the density is very low, a fourth state of matter can exist; it is called "plasma." A plasma consists of electrons and positively charged ions rather than neutral atoms, and so it has both electric and magnetic fields. (An ion is an atom that has lost one or more of its electrons.) On Earth, plasmas exist, at least temporarily, in lightning, electrical sparks, fluorescent lamps, and in the ionosphere.

In addition to the electromagnetic radiation we sense as heat and light, it is now known that the Sun emits particle radiation having a wide range of energies. The particles (or plasma) appear to come from specific regions on the Sun, some as highly energetic particles which move radially outward into interplanetary space. Some of these highly energetic particles that reach Earth's ionosphere produce auroral displays (the northern lights) and affect shortwave radio transmission by modifying the ionospheric structure.

A lower energy component of the particles is emitted from the Sun on a continuous basis, and these lower energy particles also move away from the Sun in a straight line (radially). The study of this interplanetary plasma, which has been called the solar wind, is of great concern to astronomers and other scientists for several reasons. One is that the Sun is the only star we are close to, and the emission of plasma means that it is very gradually losing matter, an important factor in stellar evolution. Another is that the plasma state of matter is difficult to study on Earth because it is hard to reproduce in the laboratory the conditions of high temperature and low density that exist naturally in the solar atmosphere and in interplanetary space.

A number of space probes and satellites have been used to investigate the properties of the interplanetary plasma. The Interplanetary Monitoring Platform (IMP) series of probes from 1963 to the present, the Orbiting Geophysical Observatory (OGO) series from 1964 to 1974, the lnternational Sun-Earth Explorer (ISEE) satellites from 1977 to the present, and the Mariner, Pioneer, and Voyager deep-space probes have all carried experiments resulting in a series of measurements of flow direction, density, velocity, and electric and magnetic fields of the solar wind.

It has been postulated, on theoretical grounds, that the magnetic field lines of the solar wind coincide with the locus of particles emitted from the Sun, and the experimental findings to date seem to support this hypothesis.

a. Determine the shape of this locus, given that the solar atmosphere from which emission takes place rotates at a constant angular velocity and that particles move outward with constant velocity in the radial direction. Assume the direction of rotation is clockwise.

CALCULUS GRAPHIC 1A.

CALCULUS GRAPHIC 1B.

CALCULUS GRAPHIC 1C.

CALCULUS GRAPHIC 1D.

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